Fixation probabilities for multi-allele Moran dynamics with weak selection

TL;DR

This paper develops a perturbative framework to analytically compute fixation probabilities in multi-allele Moran processes under weak selection, extending understanding beyond pairwise interactions.

Contribution

It introduces a systematic expansion method for fixation probabilities in multi-allele systems with arbitrary fitness functions under weak selection.

Findings

Derived fixation probabilities for three biological models

Extended analytical results to multi-allele Moran processes

Provided a framework for multi-strategy stochastic dynamics

Abstract

Fixation probabilities are essential for characterizing stochastic evolutionary dynamics, but analytical results remain limited mainly to systems with two competing types. We develop a perturbative framework to compute fixation probabilities in multi-allele Moran processes under weak selection. Exploiting the general structure of the backward Fokker-Planck operator in this regime, we show that fixation probabilities admit a systematic expansion around their neutral solution. We first introduce the framework in a general case with $M$ competing alleles and arbitrary fitness functions, and then apply it to three biologically motivated examples: a simple model of three competing alleles with a constant fitness function, a coordination game in which allele fitness increases with its frequency in the population, and a model of clonal interference between mutualistic alleles. These results extend the analytical understanding of fixation probabilities beyond pairwise interactions, establishing a framework for investigating multi-strategy stochastic evolutionary dynamics.